Peer-reviewedCC BY 4.0Nonlinear Differential Equations and Applications
On the zero-noise limit for SDE’s singular at the initial time
Abstract
We investigate the zero-noise limit for SDE's driven by Brownian motion with a divergence-free drift singular at the initial time and prove that a unique probability measure concentrated on the integral curves of the drift is selected. More precisely, we prove uniqueness of the zero-noise limit for divergence-free drifts in L1loc ((0, T ];BV(𝕋d;ℝd))∩ Lq((0,T);Lp(𝕋d;ℝd)) where p and q satisfy a Prodi-Serrin condition. The vector field constructed by Depauw lies in this class and we show that for almost every initial datum, the zero-noise limit selects a probability measure concentrated on several distinct integral curves of this vector field.
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BibTeX
@article{pitcho2026zeronoise,
title = {{On the zero-noise limit for SDE’s singular at the initial time}},
author = {Jules Pitcho},
journal = {Nonlinear Differential Equations and Applications},
year = {2026},
month = sep,
doi = {10.1007/s00030-025-01140-9},
url = {https://hal.science/hal-05756979v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - On the zero-noise limit for SDE’s singular at the initial time
AU - Jules Pitcho
PY - 2026
DA - 2026/09/21
JO - Nonlinear Differential Equations and Applications
DO - 10.1007/s00030-025-01140-9
UR - https://hal.science/hal-05756979v1
AB - We investigate the zero-noise limit for SDE's driven by Brownian motion with a divergence-free drift singular at the initial time and prove that a unique probability measure concentrated on the integral curves of the drift is selected. More precisely, we prove uniqueness of the zero-noise limit for divergence-free drifts in L^1_loc ((0, T ];BV(𝕋^d;ℝ^d))∩ L^q((0,T);L^p(𝕋^d;ℝ^d)) where p and q satisfy a Prodi-Serrin condition. The vector field constructed by Depauw lies in this class and we show that for almost every initial datum, the zero-noise limit selects a probability measure concentrated on several distinct integral curves of this vector field.
SN - 1021-9722
SN - 1420-9004
ER -